Set-Valued Analysis

, Volume 11, Issue 4, pp 323–344

Wijsman and Hit-and-Miss Topologies of Quasi-Metric Spaces

  • Jesús Rodríguez-López
  • Salvador Romaguera

DOI: 10.1023/A:1025675400451

Cite this article as:
Rodríguez-López, J. & Romaguera, S. Set-Valued Analysis (2003) 11: 323. doi:10.1023/A:1025675400451


The relationship between the Wijsman topology and (proximal) hit-and-miss topologies is studied in the realm of quasi-metric spaces. We establish the equivalence between these hypertopologies in terms of Urysohn families of sets. Our results generalize well-known theorems and provide easier proofs. In particular, we prove that for a quasi-pseudo-metrizable space (X,T) the Vietoris topology on the set P0(X) of all nonempty subsets of X is the supremum of all Wijsman topologies associated with quasi-pseudo-metrics compatible with T. We also show that for a quasi-pseudo-metric space (X,d) the Hausdorff extended quasi-pseudo-metric is compatible with the Wijsman topology on P0(X) if and only if d−1 is hereditarily precompact.

quasi-pseudo-metric(proximal) hit-and-miss topologiesWijsman topologyVietoris topologyfar Urysohn familyuniformly Urysohn familyHausdorff quasi-pseudo-metric

Copyright information

© Kluwer Academic Publishers 2003

Authors and Affiliations

  • Jesús Rodríguez-López
    • 1
  • Salvador Romaguera
    • 1
  1. 1.Escuela Politécnica Superior de Alcoy, Departamento de Matemática AplicadaUniversidad Politécnica de ValenciaAlcoy (Alicante)Spain